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This work is devoted to the numerical study of high-speed mixing layers of compressible flows. The problem under consideration has a wide range of applications in practical tasks and, despite its apparent simplicity, is quite complex in terms of modeling. Because in the mixing layer, as a result of the instability of the tangential discontinuity of velocities, the flow passes from laminar flow to turbulent mode. Therefore, the obtained numerical results of the considered problem strongly depend on the adequacy of the used turbulence models. In the presented work, this problem is studied based on the two-fluid approach to the problem of turbulence. This approach has arisen relatively recently and is developing quite rapidly. The main advantage of the two-fluid approach is that it leads to a closed system of equations, when, as is known, the long-standing Reynolds approach leads to an open system of equations. The paper presents the essence of the two-fluid approach for modeling a turbulent compressible medium and the methodology for numerical implementation of the proposed model. To obtain a stationary solution, the relaxation method and Prandtl boundary layer theory were applied, resulting in a simplified system of equations. In the considered problem, high-speed flows are mixed. Therefore, it is also necessary to model heat transfer, and the pressure cannot be considered constant, as is done for incompressible flows. In the numerical implementation, the convective terms in the hydrodynamic equations were approximated by the upwind scheme with the second order of accuracy in explicit form, and the diffusion terms in the right-hand sides of the equations were approximated by the central difference in implicit form. The sweep method was used to implement the obtained equations. The SIMPLE method was used to correct the velocity through the pressure. The paper investigates a two-liquid turbulence model with different initial flow turbulence intensities. The obtained numerical results showed that good agreement with the known experimental data is observed at the inlet turbulence intensity of $0.1 < I < 1 \%$. Data from known experiments, as well as the results of the $k − kL + J$ and LES models, are presented to demonstrate the effectiveness of the proposed turbulence model. It is demonstrated that the two-liquid model is as accurate as known modern models and more efficient in terms of computing resources.

A new flow modeling method on unstructured grid was proposed. As a basis system this method used quasi-hydro-dynamic equations. The finite volume method vas used for solving these equations. The Delaunay triangulation was used for constructing mesh. This proposed method was tested in modeling of incompressible flow through a channel with complex profile. The acquired results showed that the proposed method could be used in flow modeling in unstructured grid.

An approach to evaluation of a parametric portrait of integral equations of the theory of liquids in the RISM approximation was proposed. To obtain all associated solutions the continuation method was used. The equations reduced to a two-centered molecule model for symmetry reasons were deduced for molecular liquids. For molecular liquids, some equations were obtained which could be reduced, for symmetry reasons, to a two-center molecular model. To avoid critical points we changed the dependence of RISM-equations on reverse compressibility. The suggested method was used to perform numerical computations of methane reverse compressibility isotherms with three closures. No bifurcation of solutions was observed in the case of the partially linearized hypernetted chain closure. For other closures bifurcations of solutions were obtained and the model behavior nontypical for simple liquids was observed. In the case of Percus-Yevick closure nonphysical solutions were obtained at low temperature and density. Additional solution branch with a kink in the bifurcation point was obtained in the case of hypernetted chain closure at temperature above the critical point.

Numerical methods of obtaining collective and one-particle states were used for the quantum description of two-nuclear systems behavior at the initial stage of near-barrier heavy nuclei fusion. The collective exited states in such systems represent concordant oscillations of surfaces of spherical nuclei. The one-particle states of the external neutrons are similar to the states of valence electrons of diatomic molecules.

A mathematical model for the evolution of microbial populations during prolonged cultivation in a chemostat has been constructed. This model generalizes the sequence of the well-known mathematical models of the evolution, in which such factors of the genetic variability were taken into account as chromosomal mutations, mutations in plasmid genes, the horizontal gene transfer, the plasmid loss due to cellular division and others. Liapunov’s function for the generic model of evolution is constructed. The existence proof of bounded, positive invariant and globally attracting set in the state space of the generic mathematical model for the evolution is presented because of the application of La-Salle’s theorem. The analytic description of this set is given. Numerical methods for estimate of the number of limit sets, its location and following investigation in the mathematical models for evolution are discussed.

The method of modelling of a compact bone tissue microstructure is presented. The modelling sample is considered as set of the structural elements containing reinforcing element – osteon and a matrix. The form of structural elements is defined by distances to next osteons and directions of next osteons arrangement. Calculation of the stress and strain state of the modelling sample is carried out at tension in program complex ANSYS. Results of calculation have shown, that haversian canals are stress concentrators.

The paper presents a new dense material modeling method based on spherepolyhedra packing algorithm, describes mathematical model of spherepolyhedra and discuss the results of computation experiments on different spherepolyhedra packs. The results of experiments show convergence of proposed method. Experiments include investigations of spherepolyhedra packs with different shapes, polydisperse and oriented structures. Presented method would be applied to virtual design of dense materials composed of non-spherical particles.

This work considers the problem of optimal control galvanic process in multianode bath. The nonstationary mathematical model of galvanic process, which considers change concentrations of electrolyte components, is developed. Demonstrated rationale for the choice of the form to extremal control on example chrome galvanic process in the standard electrolyte.

This paper presents a method for obtaining fractal signals using fractal splines similar to signals generated by fractal functions. The hypothesis about the identity of discrete fractal functions and linear fractal splines is justified. There are considered the features of planning matrix calculation of cumulative fractal spline, examples of generated curves are shown.

The given work is devoted aerodynamic properties of system of the bodies which are flowed round by a supersonic stream. The question on reduction of mutual influence with increase in the size characterising scattering of elements of system is considered. The method of construction of a grid is applied to current modeling from a set of grids. One of grids, regular with rectangular cells, is responsible for an interference between bodies
and serves for the description of an external nonviscous current. Other grids are  connected with surfaces of streamline bodies and allow to describe viscous layers about streamline bodies. These grids are imposed on the first, without combination of any knots. Boundary conditions are realized through interpolation of functions on borders from one grid on another.